The Complexity of Solitaire

Abstract

Klondike is the well-known 52-card Solitaire game available on almost every computer. The problem of determining whether an n-card Klondike initial configuration can lead to a win is shown NP-complete. The problem remains NP-complete when only three suits are allowed instead of the usual four. When only two suits of opposite color are available, the problem is shown NL-hard. When the only two suits have the same color, two restrictions are shown in AC0 and in NL respectively. When a single suit is allowed, the problem drops in complexity down to AC0[3], that is, the problem is solvable by a family of constant depth unbounded fan-in {and, or, mod3}-circuits. Other cases are studied: for example, “no King” variant with an arbitrary number of suits of the same color and with an empty “pile” is NL-complete.